Three different porous substrates (with different pore sizes, $s$, and permeabilities, $K$) are used to examine their effect on the structure of boundary layer flow over them. The flow is characterised with single-point hot-wire measurements as well as planar particle image velocimetry. In order to elucidate differences in shallow and deep flows past porous substrates, foams with two different thicknesses ($h$) are used (for all three substrates). A wide range of friction Reynolds numbers ($2000< Re_\tau <13\,500$) and permeability-based Reynolds numbers ($1< Re_K< 50$) are attained. For substrates with $Re_K \sim 1$, the flow behaviour remains similar to flow over impermeable smooth walls and as such Townsend's hypothesis remains valid. Very large-scale motions are observed over permeable foams even when the $Re_K > 1$. In contrast, a substantial reduction in velocity disturbances and associated length scales is achieved for permeable foams with intermediate values of pore density and relative foam thickness ($h/s$), which affects outer-layer similarity. As permeability is increased by increasing pore size, the foam becomes sparse relative to viscous scales at high Reynolds numbers. For such foams, the flow conforms to outer-layer similarity and is more akin to flow over rough surfaces. Permeability attenuates the wavelengths associated with the outer-layer peak.

Conservation laws that relate the local time-rate-of-change of the spatial integral of a density function to the divergence of its flux through the boundaries of the integration domain provide integral constraints on the spatio-temporal development of a field. Here we show that a new type of conserved quantity exists that does not require integration over a particular domain but which holds locally at any point in the field. This is derived for the pseudo-energy density of non-divergent Rossby waves where local invariance is obtained for (i) a single plane wave, and (ii) waves produced by an impulsive point source of vorticity. The definition of pseudo-energy used here consists of a conventional kinetic part, as well as an unconventional pseudo-potential part, proposed by Buchwald (Proc. R. Soc. Lond. A, vol. 328, issue 1572, 1972, pp. 37–48). The anisotropic nature of the energy flux that appears in response to the point source further clarifies the role of the beta plane in the observed western intensification of ocean currents.

Motivated by the recent numerical results of Khalid et al. (Phys. Rev. Lett., vol. 127, 2021, 134502), we consider the large-Weissenberg-number ($W$) asymptotics of the centre mode instability in inertialess viscoelastic channel flow. The instability is of the critical layer type in the distinguished ultra-dilute limit where $W(1-\beta )=O(1)$ as $W \rightarrow \infty$ ($\beta$ is the ratio of solvent-to-total viscosity). In contrast to centre modes in the Orr–Sommerfeld equation, $1-c=O(1)$ as $W \rightarrow \infty$, where $c$ is the phase speed normalised by the centreline speed as a central ‘outer’ region is always needed to adjust the non-zero cross-stream velocity at the critical layer down to zero at the centreline. The critical layer acts as a pair of intense ‘bellows’ which blows the flow streamlines apart locally and then sucks them back together again. This compression/rarefaction amplifies the streamwise-normal polymer stress which in turn drives the streamwise flow through local polymer stresses at the critical layer. The streamwise flow energises the cross-stream flow via continuity which in turn intensifies the critical layer to close the cycle. We also treat the large-Reynolds-number ($Re$) asymptotic structure of the upper (where $1-c=O(Re^{-2/3})$) and lower branches of the $Re$–$W$ neutral curve, confirming the inferred scalings from previous numerical computations. Finally, we remark that the viscoelastic centre-mode instability was actually first observed in viscoelastic Kolmogorov flow by Boffetta et al. (J. Fluid Mech., vol. 523, 2005, pp. 161–170).

In neutrally stratified shallow water, full-depth Langmuir cells (LCs) can interact with the turbulent benthic boundary layer and, thus, influence bottom wall shear stresses. In this paper the impacts of full-depth LCs on the streamwise and spanwise wall shear stresses are systematically studied using the database obtained from wall-resolved large-eddy simulation of shallow-water Langmuir turbulence. Analyses focus on the instantaneous wall shear stress fluctuations and the joint probability density functions between the stress fluctuations and the LCs parts of the velocity fluctuations, which show that the linear superimposition effect and nonlinear modulation effect of LCs are responsible for the spanwise organized distribution of wall shear stress fluctuations. Compared with the statistics in pure shear-driven turbulence without LCs, the mean square values of wall shear stress fluctuations in shallow-water Langmuir turbulence are enhanced by the strong linear superimposition effect of LCs, while the skewness and kurtosis are reduced by the combination of the linear superimposition effect and nonlinear modulation effect of LCs. Based on the scalings of these effects, a new predictive model of wall shear stress fluctuations is proposed for shallow-water Langmuir turbulence. The proposed model can predict the spatial distribution and statistics of wall shear stress fluctuations using the LCs parts of velocity fluctuations measured above the water bottom. Owing to the persistence of the spanwise inhomogeneity of wall shear stresses induced by full-depth LCs, the new predictive model will be useful for improving the wall-layer modelling for shallow-water Langmuir turbulent flows.

We derive expressions relating the entrainment fluxes of momentum and kinetic energy, relative to the mass flux entrained into a turbulent wake exposed to a turbulent background. These expressions contain correlations between the entrainment velocity and the turbulent fluctuations within the background. We perform high-resolution, simultaneous particle image velocimetry and planar laser-induced fluorescence experiments, and observe these correlations to be negligible in the far wake, such that momentum and kinetic energy are entrained into the wake with the same relative efficiency to mass as from an idealised, non-turbulent background. This is a useful result in the context of modelling, since the entrainment hypothesis (Turner, J. Fluid Mech., vol. 173, 1986, pp. 431–471) can still be used to model the entrainment of momentum and kinetic energy. Nevertheless, the entrainment rate of mass is shown to vary spatially, and with the specific nature of the background turbulence, so this in turn drives a spatial/background-turbulence-specific entrainment rate of momentum/kinetic energy. Contrastingly, in the near wake, whilst momentum is entrained from a turbulent background with the same relative efficiency to mass as for an idealised non-turbulent background, this is not the case for kinetic energy. Owing to the sum of multiple positive, small-valued correlations between the fluctuations in the background and the entrainment velocity, kinetic energy is entrained more efficiently than in the idealised case. This includes entrainment from a non-turbulent background, where small correlations are observed between the irrotational background fluctuations and the entrainment velocity. Evidence is also presented that the entrainment velocity scales with the Kolmogorov velocity scale when the background is turbulent.

The volume filtering of the governing equations of a particle-laden flow allows for simulating the fluid phase as a continuum, and accounting for the momentum exchange between the fluid and the particles by adding a source term in the fluid momentum equations. The volume filtering of the Navier–Stokes equations allows consideration of the effect that particles have on the fluid without further assumptions, but closures arise of which the implications are not fully understood. It is common to either neglect these closures or model them using assumptions of which the implications are unclear. In the present paper, we carefully study every closure in the volume-filtered fluid momentum equation and investigate their impact on the momentum and energy transfer dependent on the filtering characteristics. We provide an analytical expression for the viscous closure that arises because the filter and spatial derivative in the viscous term do not commute. An analytical expression for the regularization of the particle momentum source of a single sphere in the Stokes regime is derived. Furthermore, we propose a model for the subfilter stress tensor, which originates from filtering the advective term. The model for the subfilter stress tensor is shown to agree well with the subfilter stress tensor for small filter widths relative to the size of the particle. We show that, in contrast to common practice, the subfilter stress tensor requires modelling and should not be neglected. For filter widths comparable to the particle size, we find that the commonly applied Gaussian regularization of the particle momentum source is a poor approximation of the spatial distribution of the particle momentum source, but for larger filter widths, the spatial distribution approaches a Gaussian. Furthermore, we propose a modified advective term in the volume-filtered momentum equation that consistently circumvents the common stability issues observed at locally small fluid volume fractions. Finally, we propose a generally applicable form of the volume-filtered momentum equation and its closures based on clear and well-founded assumptions. Based on the new findings, guidelines for point-particle simulations and the filter width with respect to the particle size and fluid mesh spacing are proposed.

Experimental and theoretical studies on millimetre-sized droplets suggest that at low Reynolds number the difference between the drag force on a circulating water droplet and that on a rigid sphere is very small (less than 1 %) (LeClair et al., J. Atmos. Sci., vol. 29, 1972, pp. 728–740). While the drag force on a spherical liquid droplet at high viscosity ratios (of the liquid to the gas), is approximately the same as that on a rigid sphere of the same size, the other quantities of interest (e.g. the temperature) in the case of a rarefied gas flow over a liquid droplet differ from the same quantities in the case of a rarefied gas flow over a rigid sphere. The goal of this article is to study the effects of internal motion within a spherical microdroplet/nanodroplet – such that its diameter is comparable to the mean free path of the surrounding gas – on the drag force and its overall dynamics. To this end, the problem of a slow rarefied gas flowing over an incompressible liquid droplet is investigated analytically by considering the internal motion of the liquid inside the droplet and also by accounting for kinetic effects in the gas. Detailed results for different values of the Knudsen number, the ratio of the thermal conductivities and the ratio of viscosities are presented for the pressure and temperature profiles inside and outside the liquid droplet. The results for the drag force obtained in the present work are in good agreement with the theoretical and experimental results existing in the literature.

A series of laboratory experiments are reported in which a continuous stream of bubbles rise from a small source at the base of a tank of water. Using different nozzles, bubble sizes $d$ ranging from 1.2 to 11.6 mm were produced for a number of gas volume fluxes, $Q_b$, ranging between 1.1 and $21.1\times 10^{-6}\ {\rm m}^3\ {\rm s}^{-1}$. Within a small distance from the source, the slip speed of these bubbles exceeds the speed of the equivalent single-phase plume with the same buoyancy flux, leading to formation of what we refer to as the ‘slip plume’ regime. Through a combination of high-speed photography, coupled with flow visualisation in the plume and the ambient fluid using dye, we find that the bubble speed and the fluid speed remain nearly constant with height, with the maximum fluid speed being of order $0.30\pm 0.03$ of the bubble speed. Using the filling box method, we also find that the net fluid volume flux in the slip plume increases linearly with distance from the source at a rate $Q_l = \lambda Bz/v_s^2$, where $B$ is the buoyancy flux of the gas, $v_s$ the rise speed of the gas bubbles, $z$ the distance above the source and $\lambda$ is a constant related to the dimensionless volume of fluid in the wake of each bubble. This slip-dominated flow regime can be understood in terms of kinetic energy imparted to the fluid as the bubbles rise and release potential energy. Further experiments with particle-laden plumes illustrate similar scalings for the volume flux in a particle-driven slip plume once the slip speed of the particles exceeds the bulk speed of the equivalent single-phase buoyant plume with the same buoyancy flux. Near the source the slip speed may be smaller than the plume speed, and the flow follows the classical model for a turbulent buoyant plume, with the transition to the slip regime occurring at a distance $z^*\approx (32\pm 5)\lambda ^{3/2} B/v_s^3$ from the source, where the dimensionless parameter $\lambda$ relates to the dimensionless volume of the fluid wake, which we find varies with the Reynolds number of the particles.

Droplet spreading is ubiquitous and plays a significant role in liquid-based energy systems, thermal management devices and microfluidics. While the spreading of non-volatile droplets is quantitatively understood, the spreading and flow transition in volatile droplets remains elusive due to the complexity added by interfacial phase change and non-equilibrium thermal transport. Here we show, using both mathematical modelling and experiments, that the wetting dynamics of volatile droplets can be scaled by the spatial–temporal interplay between capillary, evaporation and thermal Marangoni effects. We elucidate and quantify these complex interactions using phase diagrams based on systematic theoretical and experimental investigations. A spreading law of evaporative droplets is derived by extending Tanner's law (valid for non-volatile liquids) to a full range of liquids with saturation vapour pressure spanning from $10^1$ to $10^4$ Pa and on substrates with thermal conductivity from $10^{-1}$ to $10^3\ {\rm W}\ {\rm m}^{-1}\ {\rm K}^{-1}$. In addition to its importance in fluid-based industries, the conclusions also enable a unifying explanation to a series of individual works including the criterion of flow reversal and the state of dynamic wetting, making it possible to control liquid transport in diverse application scenarios.

Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [In Vladimir I. Arnold – Collected Works (ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain.

Time-varying flow separation on an accelerating prolate spheroid has been studied at various angles of incidence. Instantaneous pressure and scanning stereoscopic particle image velocimetry were used to shed light on the evolution of cross-flow structures for the Reynolds number ($Re$) range of $1.0\times 10^6\leq Re \leq 1.5\times 10^6$. The movement of separation lines is examined for various model accelerations to investigate on the interplay between acceleration and flow separation. The results demonstrate that for axial accelerations, the streamwise pressure distribution in the rear part of the prolate spheroid switches from an adverse to a favourable pressure gradient. At the same time, the circumferential adverse pressure gradient present during steady motion vanishes during said accelerations. In contrast, both streamwise and circumferential adverse pressure gradients strengthen when the model is axially decelerated. These dynamic pressure distributions influence the location of the separation line, which in turn moves closer to the model meridian during accelerations while moving outwards during decelerations. The streamwise vorticity distribution and the streamwise circulation both show how the separation-line position impacts the vortex formation. A high-vorticity region near the model surface is established during acceleration. In contrast, a decelerating model leads to transport of high-vorticity fluid into the outer area of the cross-flow separation. We further assess the memory effects following the near-impulsive velocity changes. The cross-flow retains the memory of moving separation lines shortly after the acceleration. However, the separation recovers quickly to a steady state.

The present study investigates the profiles of statistically axisymmetric turbulent jets with arbitrary buoyancy. Analytical expressions for the shape of the radial velocity, Reynolds stress and radial scalar flux profiles are derived from the governing equations by assuming self-similar Gaussian mean velocity and scalar profiles. Previously these have only been derived for the special cases of pure jets and plumes, whereas the present study generalises them to arbitrary buoyancies. These are then used to derive analytical expressions for the turbulent Schmidt/Prandtl numbers, which, along with the mean profiles, are shown to give predictions in agreement with existing literature.

We propose a semi-analytical solution for the advection–diffusion equation in cylindrical domains, with an aim towards extracting blood flow rates from contrast variations in a coronary computed tomography angiography image. The solution proposed in this work, in contrast with existing methods, which only consider advection, incorporates both radial velocity variation and diffusion. By means of a Galerkin approach using Bessel functions, a solution for a three-dimensional concentration field at a single time point is obtained after a Laplace transformation. This semi-analytical solution forms the basis for a novel advection–diffusion flow estimation (ADFE) method. The ADFE is derived, validated through numerical spectral-element method computations, and shown to exhibit improved accuracy against the state-of-the-art method for image-based blood flow extraction.

We investigate the retraction of a circular thin film coated with insoluble surfactants, which is punctured at its centre. We assume that the surface pressure of the liquid–gas interface is related to the number density of surfactants through a linear equation of state, which is characterized by a single parameter: the Gibbs dilation modulus. To solve the governing equations and track the deformation of the domain, we use the finite element method with an arbitrary Lagrangian–Eulerian approach where the film surface is sharp. Our simulations show that the surface elasticity introduced by the surfactants slows down the retraction and introduces oscillations at early times. In agreement with previous experiments and theoretical analysis, we find that the presence of surfactants introduces perturbations of the film thickness over progressively larger distances as the surface elasticity increases. The surface perturbations travel faster than the retracting edge of the film at a speed proportional to the Gibbs modulus. For large values of the Gibbs modulus, the interface behaviour approaches that of an incompressible two-dimensional solid. Our analysis sheds light on the effect of insoluble surfactants on the opening of a circular hole in a thin film and can be extended to investigate the onset of surface cracks and fractures.